2015/07/06 by Christos Pelekis, Pelekis, Christos
Computer Science · Mathematics · #05Dxx #51Fxx #52Cxx #Combinatorics #Computational Geometry (cs.CG) #Computer science #Discrete mathematics #Epistemology #FOS: Computer and information sciences #Lebesgue integration #Lebesgue measure #Limits and Structures in Graph Theory #Mathematical Approximation and Integration #Mathematics #Measure (data warehouse) #Pairwise comparison #Philosophy #Point processes and geometric inequalities #Property (philosophy) #Set (abstract data type) #Statistics #cs.CG #msc:05Dxx #msc:51Fxx #msc:52Cxx
paper · pdf · doi:10.48550/arxiv.1507.01631
15 pages, 1 figure, Minor typos corrected
openalex publication_date 2015/07/06 · arxiv created 2015/08/04 · arxiv updated 2015/08/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix positive integers a and b such that a> b≥ 2 and a positive real δ>0. Let S be a planar set of diameter δ having the following property: for every a points in S, at least b of them have pairwise distances that are all less than or equal to 2. What is the maximum Lebesgue measure of S? In this paper we investigate this problem. We discuss the, devious, motivation that leads to its formulation and provide upper bounds on the Lebesgue measure of S. Our main result is based on a generalisation of a theorem that is due to Heinrich Jung. In certain instances we are able to find the extremal set but the general case seems elusive.